Whether the ing-form of the verb is a gerund or a participle is clear to me.
I know that these are not objects of the verbs 'go' and 'come', so they may be participle.
If they are participle then they must modify some noun or pronoun and the only pronoun in these sentences are 'she'.

Linguists are not in perfect agreement about these 'go/come -ing' expressions.

Some would consider the e.g. 'go shopping' to be an abridged form of 'go for (the purpose of) shopping' with obligatory prepositional ellipsis, making 'shopping' a gerund. Many, however, would happily accept it as a participle referring back to the subject.